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From the perspective of the statistical fluctuation theory, we explore therole of the thermodynamic geometries and vacuum (in)stability properties forthe topological Einstein-Yang-Mills black holes. In this paper, from theperspective of the state-space surface and chemical Wienhold surface, weprovide the criteria for the local and global statistical stability of anensemble of topological Einstein-Yang-Mills black holes in arbitrary spacetimedimensions $D\ge 5$. Finally, as per the formulations of the thermodynamicgeometry, we offer a parametric account of the statistical consequences in boththe local and global fluctuation regimes of the topological Einstein-Yang-Millsblack holes.

Thermodynamic Geometry and Topological Einstein-Yang-Mills Black Holes